Say you are selecting $n$ observations from a complex survey of $N$ individuals to create an analytical sample of relevant observations; and that you intend to fit a binomial multivariate logistic regression to the data, where:

$$ Y = \beta_0 + \beta_1 D + \beta_2 X + \beta_3 Z + \epsilon $$

Say that for the covariate $Z$ included as control, there are $K$ missing observations. They are not missing completely at random and $K > 0.1n$.

How, mathematically, can you comparatively assess the sensitivity of your model coefficients and parameter to a choice of either dropping the K rows (listwise deletion) or omitting the Z feature?

(Related post: Missing data in a logistic regression analysis ; doesn't directly answer this question.)

Say you are selecting $n$ observations from a complex survey of $N$ individuals to create an analytical sample of relevant observations; and that you intend to fit a binomial multivariate logistic regression to the data, where:

$$ Y = \beta_0 + \beta_1 D + \beta_2 X + \beta_3 Z + \epsilon $$

Say that for the covariate $Z$ included as control, there are $K$ missing observations. They are not missing completely at random and $K > 0.1n$.

How, mathematically, can you comparatively assess the sensitivity of your model coefficients and parameter to a choice of either dropping the K rows (listwise deletion) or omitting the Z feature?

(Related post: Missing data in a logistic regression analysis ; doesn't directly answer this question.)

Edit for clarification: Indeed you may be better off proceeding with multiple imputation; but here I am interested in understanding the consequences of the above choice in cases where a missing at random assumption is not necessarily plausible.